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On the approximability of Dodgson and Young elections
, 2008
"... The voting rules proposed by Dodgson and Young are both designed to find the alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorith ..."
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Cited by 27 (10 self)
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The voting rules proposed by Dodgson and Young are both designed to find the alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: an LPbased randomized rounding algorithm and a deterministic greedy algorithm, both of which yield an O(log m) approximation ratio, where m is the number of alternatives; we observe that this result is asymptotically optimal, and further prove that our greedy algorithm is optimal up to a factor of 2, unless problems in N P have quasipolynomial time algorithms. Although the greedy algorithm is computationally superior, we argue that
Socially Desirable Approximations for Dodgson’s Voting Rule ∗ ABSTRACT
"... voting rule that today bears his name. Although Dodgson’s rule is one of the most wellstudied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is N Phard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice p ..."
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Cited by 20 (4 self)
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voting rule that today bears his name. Although Dodgson’s rule is one of the most wellstudied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is N Phard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice point of view—it fails basic social choice desiderata such as monotonicity and homogeneity. In a previous paper [Caragiannis et al., SODA 2009] we have asked whether there are approximation algorithms for Dodgson’s rule that are monotonic or homogeneous. In this paper we give definitive answers to these questions. We design a monotonic exponentialtime algorithm that yields a 2approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomialtime O(log m)approximation algorithm (where
A complexityofstrategicbehavior comparison between Schulze’s rule and ranked pairs
 In Proc. of 26th AAAI Conference on AI
, 2012
"... Schulze’s rule and ranked pairs are two Condorcet methods that both satisfy many natural axiomatic properties. Schulze’s rule is used in the elections of many organizations, including ..."
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Cited by 16 (0 self)
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Schulze’s rule and ranked pairs are two Condorcet methods that both satisfy many natural axiomatic properties. Schulze’s rule is used in the elections of many organizations, including
Possible Winners When New Alternatives Join: New Results Coming Up!
"... In a voting system, sometimes multiple new alternatives will join the election after the voters’ preferences over the initial alternatives have been revealed. Computing whether a given alternative can be a cowinner when multiple new alternatives join the election is called the possible cowinner wi ..."
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Cited by 12 (5 self)
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In a voting system, sometimes multiple new alternatives will join the election after the voters’ preferences over the initial alternatives have been revealed. Computing whether a given alternative can be a cowinner when multiple new alternatives join the election is called the possible cowinner with new alternatives (PcWNA) problem, introduced by Chevaleyre et al. [4, 5]. In this paper, we show that the PcWNA problems are NPcomplete for the Bucklin, Copeland0, and Simpson (a.k.a. maximin) rule, even when the number of new alternatives is no more than a constant. We also show that the PcWNA problem can be solved in polynomial time for plurality with runoff. For the approval rule, we define three different ways to extend a linear order with new alternatives, and characterize the computational complexity of the PcWNA problem for each of them. 1
New Candidates Welcome! Possible Winners with respect to the Addition of New Candidates
, 2010
"... In some voting contexts, some new candidates may show up in the course of the process. In this case, we may want to determine which of the initial candidates are possible winners, given that a fixed number k of new candidates will be added. We give a computational study of the latter problem, focusi ..."
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Cited by 11 (5 self)
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In some voting contexts, some new candidates may show up in the course of the process. In this case, we may want to determine which of the initial candidates are possible winners, given that a fixed number k of new candidates will be added. We give a computational study of the latter problem, focusing on scoring rules, and we give a formal comparison with related problems such as control via adding candidates or cloning.
Proportional Representation as Resource Allocation: Approximability Results
, 2012
"... We model Monroe’s and Chamberlin and Courant’s multiwinner voting systems as a certain resource allocation problem. We show that for many restricted variants of this problem, under standard complexitytheoretic assumptions, there are no constantfactor approximation algorithms. Yet, we also show case ..."
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Cited by 8 (1 self)
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We model Monroe’s and Chamberlin and Courant’s multiwinner voting systems as a certain resource allocation problem. We show that for many restricted variants of this problem, under standard complexitytheoretic assumptions, there are no constantfactor approximation algorithms. Yet, we also show cases where good approximation algorithms exist (briefly put, these variants correspond to optimizing total voter satisfaction under Borda scores, within Monroe’s and Chamberlin and Courant’s voting systems). 1
Search versus Decision for Election Manipulation Problems
, 2013
"... Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for man ..."
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Cited by 8 (6 self)
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Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.
Fully Proportional Representation as Resource Allocation: Approximability Results
 PROCEEDINGS OF THE TWENTYTHIRD INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2013
"... We study the complexity of (approximate) winner determination under Monroe’s and ChamberlinCourant’s multiwinner voting rules, where we focus on the total (dis)satisfaction of the voters (the utilitarian case) or the (dis)satisfaction of the worstoff voter (the egalitarian case). We show good appro ..."
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Cited by 8 (6 self)
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We study the complexity of (approximate) winner determination under Monroe’s and ChamberlinCourant’s multiwinner voting rules, where we focus on the total (dis)satisfaction of the voters (the utilitarian case) or the (dis)satisfaction of the worstoff voter (the egalitarian case). We show good approximation algorithms for the satisfactionbased utilitarian cases, and inapproximability results for the remaining settings.
How Many Vote Operations Are Needed to Manipulate a Voting System?
, 2012
"... In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate n votes i.i.d. according to a distribution π, and let n go to infinity, then for any ɛ > 0, ..."
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Cited by 7 (3 self)
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In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate n votes i.i.d. according to a distribution π, and let n go to infinity, then for any ɛ > 0, with probability at least 1 − ɛ, the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) Θ ( √ n), (3) Θ(n), and (4) ∞. This theorem holds for any set of vote operations, any individual vote distribution π, and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also show that many wellstudied types of strategic behavior fall under our framework, including (but not limited to) constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size. Therefore, our main theorem naturally applies to these problems.